2 Clarified the question

Maybe this is a silly question (or not even a question), but I was wondering whether the cotangent bundle of a submanifold is somehow canonically related to the cotangent bundle of the ambient space. To be more precise:
Let $N$ be a manifold and $\iota:M \hookrightarrow N$ be an embedded (immersed) submanifold. Is the cotangent bundle $T^\ast M$ somehow canonical related to the cotangent bundle $T^\ast N$. Canonical means, without choosing a metric on $N$. The choice of a metric gives an isomorphism of $TN$ and $T^\ast N$ and therefore a "relation", since the tangent bundle of the submanifold $M$ can be viewed in a natural way as a subspace of the tangent bundle of the ambient space $N$ ($\iota$ induces an injective linear map at each point $\iota_\ast : T_pM \rightarrow T_pN$). I think this is not true for the cotangent space (without a metric)
Moreover, the cotangent bundle $T^\ast N$ of a manifold $N$ is a kind of "prototype" of a symplectic manifold. The symplectic structure on $T^\ast N$ is given by $\omega_{T^\ast N} = -d\lambda$, where $\lambda$ is the Liouville form on the cotangent bundle. (tautological one-form, canonical one-form, symplectic potential or however you want). The cotangent bundle of the submanifold $T^\ast M$ inherits in the same way a canonical symplectic structure. So, is there a relation between $T^\ast N$ and $T^\ast M$ respecting the canonical symplectic structures. (I think the isomorphism given by a metric is respecting (relating) these structures, or am I wrong?) As I said, this question is perhaps strange, but the canonical existence of the symplectic structure on the cotangent bundle is "quite strong". For example:
A given diffeomorphism $f:X \rightarrow Y$ induces a canonical symplectomorphism $T^\ast f : T^\ast Y \rightarrow T^\ast X$ (this can be proved by the special "pullback cancellation" property of the Liouville form). So in the case of a diffeomorphism the symplectic structures are "the same". Ok, a diffeomorphism has more structure than an embedding, but perhaps there is a similar relation between $T^\ast M$ and $T^\ast N$?

EDIT: Sorry fot the confusion, but Kevins post is exactly a reformulation of the problem, I'm interested in. To clarify things: with the notation of Kevin's post:

When (or whether) are the pulled back symplectic structures the same ? Under what circumstances holds $a^\ast \omega_{T^\ast N} = b^\ast \omega_{T^\ast N}$

I think this isn't true for any submanifold $M \subset N$, but what is a nice counterexample? Is it true for more restricted submanifolds as for example embedded submanifolds which are not just homoeomorphisms onto its image, but diffeomorphisms (perhaps here the answer is yes, using the diffeomorphism remark above?)?

1

# Cotangent bundle of a submanifold

Maybe this is a silly question (or not even a question), but I was wondering whether the cotangent bundle of a submanifold is somehow canonically related to the cotangent bundle of the ambient space. To be more precise:
Let $N$ be a manifold and $\iota:M \hookrightarrow N$ be an embedded (immersed) submanifold. Is the cotangent bundle $T^\ast M$ somehow canonical related to the cotangent bundle $T^\ast N$. Canonical means, without choosing a metric on $N$. The choice of a metric gives an isomorphism of $TN$ and $T^\ast N$ and therefore a "relation", since the tangent bundle of the submanifold $M$ can be viewed in a natural way as a subspace of the tangent bundle of the ambient space $N$ ($\iota$ induces an injective linear map at each point $\iota_\ast : T_pM \rightarrow T_pN$). I think this is not true for the cotangent space (without a metric)
Moreover, the cotangent bundle $T^\ast N$ of a manifold $N$ is a kind of "prototype" of a symplectic manifold. The symplectic structure on $T^\ast N$ is given by $\omega_{T^\ast N} = -d\lambda$, where $\lambda$ is the Liouville form on the cotangent bundle. (tautological one-form, canonical one-form, symplectic potential or however you want). The cotangent bundle of the submanifold $T^\ast M$ inherits in the same way a canonical symplectic structure. So, is there a relation between $T^\ast N$ and $T^\ast M$ respecting the canonical symplectic structures. (I think the isomorphism given by a metric is respecting (relating) these structures, or am I wrong?) As I said, this question is perhaps strange, but the canonical existence of the symplectic structure on the cotangent bundle is "quite strong". For example:
A given diffeomorphism $f:X \rightarrow Y$ induces a canonical symplectomorphism $T^\ast f : T^\ast Y \rightarrow T^\ast X$ (this can be proved by the special "pullback cancellation" property of the Liouville form). So in the case of a diffeomorphism the symplectic structures are "the same". Ok, a diffeomorphism has more structure than an embedding, but perhaps there is a similar relation between $T^\ast M$ and $T^\ast N$?